Week 11 · Two Equations at Once
Puzzle Day
Puzzle Day: the crossing strings
// Where the lines cross
⏱ about 20 min
Puzzle Day: The Crossing Strings
Saturday the floor grid is taped out and the family gathers around it in their shoes.
"Two cards," Rocket says. "y = x + 2 and y = 2x. Guess the crossing before the strings go down."
Guesses fly. "(1, 3)." "(2, 4)." "(3, 5)."
"What do you notice when you test (2, 4) in both?" Raven asks.
"2 + 2 is 4, and 2 × 2 is 4. Both true!"
Nova hovers above the grid, her light dimming to a hint. "Now lay the strings and see."
The red string crosses the blue right over (2, 4). Everyone cheers.
"Next pair," Rocket says. "This time you write the equations and we guess."
- A system is two equations that must both be true. Its solution is the crossing point of the two lines.
- Solve by inspection, by graphing or by substitution.
- Same slope and different intercepts: parallel, no solution. The same line twice: infinitely many.
- A two-unknown puzzle needs two facts, one equation each.
- Always check the pair in both equations and in the story.
◇ FAMILY CROSSING STRINGS
Tape a grid on the floor. One person writes two equations in y = mx + b form on a card.
Everyone else guesses the crossing point. Then lay the strings and see who was closest.
Check the winner's point in both equations. Trade roles and build a new pair.
For a twist, write two parallel lines and watch everyone hunt for a crossing that never comes.
Everyone else guesses the crossing point. Then lay the strings and see who was closest.
Check the winner's point in both equations. Trade roles and build a new pair.
For a twist, write two parallel lines and watch everyone hunt for a crossing that never comes.
For a grown-up
Tape every edge of the grid flat and keep shoes on. Strings are taped at both ends and never wrapped around anyone.
Scissors for cutting string stay with a grown-up.
A guess that is off is a good moment: test it in both equations and see which one it fails.
Scissors for cutting string stay with a grown-up.
A guess that is off is a good moment: test it in both equations and see which one it fails.
PUZZLE DAY CHECK
- Read the question.
- Tap your answer.
The family cards say y = x + 2 and y = 2x. Where do the strings cross?Which point makes both y = x + 1 and y = 3x - 3 true?
The twist cards say y = x + 2 and y = x + 5. How many solutions does the system have?
The family cards were y = x + 2 and y = 2x. What is the y of the crossing point? Type the number.
Write 2x in place of y: x + 2 = 2x. Solve for x, then find y.
WHY THIS EXERCISEA guess becomes an answer only when it passes both equations.
Draw the family grid with y = x + 2 and y = 2x. Mark the crossing and one guess that missed.